I hesitate to ask G-d for anything because I feel as though it is not my place to ask. I feel as though it is solely my responsibility to provide for myself using the means that G-d has already blessed me with, and that I should be grateful G-d gave me anything at all to begin with because he was under no obligation to give me life or any of the luxuries I've enjoyed while alive.

Am I wrong for feeling this way? I have felt tempted to ask G-d for help during difficult times, but I refrained from doing so because I thought I would be out of line. Instead, I simply thanked him for all that he has already given me and did what I could to provide for myself.

If it is permissible to ask G-d for help, is it advisable? Under what circumstances can one ask him for help? What can one ask him for?

There are things that I need and things that I want, but so far I have asked G-d for none of them and just assumed that if he wanted me to have them, he would have given me the ability to obtain them on my own. Am I wrong?

Rav Moshe Feinstein in Igros Moshe OC II:25 says it's even a mitzvah (so to speak) for non Jews to pray for their needs, as it is a function of faith in Hashem, which is a prerequisite to observing the seven Noahides mitzvos.

There are various aggadic passages that indicate Hashem wants us to pray to Him (what the reason is is another question). For example, Yevamos 64a says

Rabbi Yitzḥak said: For what reason were our forefathers initially infertile? Because the Holy One, Blessed be He, desires the prayers of the righteous (and He therefore wanted them to pray for children).

From all these sources we see it is not only appropriate but in many cases incumbent upon us to pray to Hashem for our needs.

Regarding your other question, the Mishnah in Berachos 9:3 says what types of prayers are inappropriate.

He who prays over what has already happened, this prayer is in vain. How? If his wife was pregnant and he said, “May it be your will that my wife give birth to a boy," this prayer is in vain. If he was coming on the way and heard the sound of screaming in the city, and he said, “May it be your will that these are not the children of my house," this is a prayer in vain.

Before there was a set liturgy for prayer each person would pray as they could and as they needed. The "siddur" only came about because of the dispersion of Jews, forgetting Hebrew. But from a basic reading of the rambam it sounds like the basic obligation to ask hashem for your needs still stands even after you pray from the siddur of you particular need hasn't been mentioned. (Rambam hilchos tefillah perek 1, 1-4)